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The triangles below are similar. The ratio of the lengths of corresponding sides is 49 . a. What is the ratio (smaller to larger) of the perimeters? b. What is the ratio (smaller to larger) of the areas? The image is of two similar triangles with proportionate side of each marked as 4 in. and 9 in. respectively.

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temdan2001

Answer:

a.) 4 : 9

b.) 16 : 81

Explanation:

a.) Ratio of perimeter will still be

4 : 9

b.) The two similar triangles  ΔABC and ΔDEF as shown from the attached figures. That is

AB/DE = AC/DF = BC/EF

It is apparent that ΔAPB and ΔDEQ are also similar as all respective angles are equal. Hence,

AB/DE = AP/DQ = BP/EQ

Moreso ΔABC = 1/2 × BC × AP  and   ΔDEF = 1/2 × EF × DQ and

ΔAPB/ΔDEQ = BC/EF × AP/DQ

But 

AP/DQ = AB/DE = BC/EF

 and hence

ΔAPB/ΔDEQ = BC/EF × BC/EF

= BC^2/EF^2  and as

BC/EF = AC/DF = AB/DE

ΔAPB/ΔDEQ = AC^2/DF^2 = BC^2/EF^2

= AB^2/DE^2

Hence if sides of two similar triangles are in the ratio a:b, their areas are in the proportion a^2:b^2

As in given case sides are in the ratio of 4:9.

ratio of their areas is 4^2:9^2 or 16:81.

Ver imagen temdan2001
Lanuel

The ratio (smaller to larger) of the perimeters of this similar triangle is equal to 4:9 and 16:81 for their areas.

Given the following data:

  • Ratio = 4:9
  • Side length of smaller triangle = 4 inches.
  • Side length of larger triangle = 9 inches.

What is a triangle?

A triangle can be defined as a two-dimensional shape that comprises three (3) sides, three (3) vertices and three (3) angles.

How to calculate the perimeter of a triangle.

Mathematically, the perimeter of a triangle is given by this formula:

[tex]P=A+B+C[/tex]

Where:

  • A, B, and C are length of sides.

What is scale factor?

Scale factor can be defined as the ratio of two (2) corresponding length of sides in two similar geometric figures such as equilateral triangles.

In Geometry, if the scale factor of two (2) similar geometric figures is given as a:b, then;

  1. The ratio of their perimeters is equal to a:b.
  2. The ratio of their areas is equal to [tex]a^2:b^2[/tex].

In conclusion, the ratio (smaller to larger) of the perimeters of this similar triangle is equal to 4:9. Also, the ratio (smaller to larger) of their areas is equal to [tex]4^2 :9^2=[/tex] 16:81.

Read more on perimeter of a triangle here: https://brainly.com/question/24382052

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